On a new unified integral

In the present paper we derive a unified new integral whose integrand contains products of FoxH-function and a general class of polynomials having general arguments. A large number of integrals involving various simpler functions follow as special cases of this integral.     

Double convolution integral equations involving a general class of multivariable polynomials and the multivariableH-functions

In this paper we have solved a double convolution integral equation whose kernel involves the product of theH-functions of several variables and a general class of multivariable polynomials. Due to general nature of the kernel, we can obtain from it, solutions of a large number of double and single convolution integral equations involving products of several classical orthogonal polynomials and simpler functions. We have also obtained here solutions of two double convolution integral equations as special cases of our main result. Exact reference of three known results, which are obtainable as particular cases of one of these special cases, have also been included.     

Fractional integral formulae involving a general class of polynomials and the multivariableH-function

We obtain two fractional integral formulae involving a general class of polynomials and the multivariableH-function. On account of the most general nature of the polynomials and the multivaribleH-function involved herein, our findings provide interesting unifications and extensions of a number of (known and new) results. We have mentioned here only two such results.     

Multidimensional modified fractional calculus operators involving a general class of polynomials

In the present work, we introduce and study essentially a class of multi-dimensional modified fractional calculus operators involving a general class of polynomials in the kernel. These operators are considered in the space of functionsM _γ (R ₊ ⁿ ). Some mapping properties and fractional differential formulas are obtained. Also images of some elementary and special functions are established.     

A note on multidimensional modified fractional calculus operators

In the present investigation some new formulas giving the images under multidimensional modified fractional operators of the celebratedH-function of Fox [Trans. Am. Math. Soc. 98 (1961) 395–429] are obtained. Special cases are briefly pointed out and the results are also studied on general spaces of functionsM _γ(R ₊ ⁿ )     

Sharp bounds for singular values of fractional integral operators

From the results of Dostanic [M.R. Dostanic, Asymptotic behavior of the singular values of fractional integral operators, J. Math. Anal. Appl. 175 (1993) 380-391] and V? and Gorenflo [Kim Tuan V?, R. Gorenflo, Singular values of fractional and Volterra integral operators, in: Inverse Problems and Applications to Geophysics, Industry, Medicine and Technology, Ho Chi Minh City, 1995, Ho Chi Minh City Math. Soc., Ho Chi Minh City, 1995, pp. 174-185] it is known that the jth singular value of the fractional integral operator of order α>0 is approximately (πj)^−α for all large j. In this note we refine this result by obtaining sharp bounds for the singular values and use these bounds to show that the jth singular value is (πj)^−α[1+O(j⁻¹)].     

Certain fractional derivative formulae involving the product of a general class of polynomials and the multivariableH-function

In the present paper, we obtain three unified fractional derivative formulae (FDF). The first involves the product of a general class of polynomials and the multivariableH-function. The second involves the product of a general class of polynomials and two multivariableH-functions and has been obtained with the help of the generalized Leibniz rule for fractional derivatives. The last FDF also involves the product of a general class of polynomials and the multivariableH-function but it is obtained by the application of the first FDF twice and it involves two independent variables instead of one. The polynomials and the functions involved in all our fractional derivative formulae as well as their arguments which are of the typex ^ρ Π _i=1 ^s (x ^t _i +α _i ) ^σ _i are quite general in nature. These formulae, besides being of very general character have been put in a compact form avoiding the occurrence of infinite series and thus making them useful in applications. Our findings provide interesting unifications and extensions of a number of (new and known) results. For the sake of illustration, we give here exact references to the results (in essence) of five research papers [2, 3,10, 12, 13] that follow as particular cases of our findings. In the end, we record a new fractional derivative formula involving the product of the Hermite polynomials, the Laguerre polynomials and the product ofr different Whittaker functions as a simple special case of our first formula.     

On fractional maximal function and fractional integral on the Laguerre hypergroup

Let be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator and the fractional integral operator on the Laguerre hypergroup from the spaces to the spaces and from the spaces to the weak spaces .     

Modulation space estimates for the fractional integral operators

We obtain the boundedness for the fractional integral operators from the modulation Hardy space μp,q to the modulation Hardy space μr,q for all 0 < p < ∞. The result is an extension of the known result for the case 1 < p < ∞ and it contains a larger range of r than those in the classical result of the Lp → Lr boundedness in the Lebesgue spaces. We also obtain some estimates on the modulation spaces for the bilinear fractional operators.     

A theorem concerning a product of two general classes of polynomials and the multivariableH-function

A theorem concerning a product of two general classes of polynomials and the multivariableH-function is established. Certain integrals and expansion formulae have also been derived by the application of this theorem. This general theorem yields a number of new, interesting and useful theorems, integrals and expansion formulae as its particular cases.     

A formula for the powers of fractional order operators

We derive none some explicit formula for the power of fractional order (differential and integral) operators.     

Weighted estimates for bilinear fractional integral operators

Moen (2016) proved weighted estimates for the bilinear fractional integrals where . We improve his results when and consider the case . As a corollary we obtain a bilinear Stein–Weiss inequality where .     

Modular estimates of fractional integral operators and k-plane transforms

The modular estimates for the fractional integral operators and the k-plane transforms are obtained in this paper. These estimates are obtained by using the modular estimates of Hardy operators and the modular interpolation theorem.     

On Urysohn-Volterra fractional quadratic integral equations

In this paper the authors study a fractional quadratic integral equation of Urysohn-Volterra type. They show that the integral equation has at least one monotonic solution in the Banach space of all real functions defined and continuous on the interval $[0,1]$. The main tools in the proof are a fixed point theorem due to Darbo and a monotonicity measure of noncompactness.     

Weak estimates for commutators of fractional integral operators

By introducing a kind of maximal operator of the fractional order associated with the mean Luxemburg norm and using the technique of the sharp function, the weak type LlogL estimates for the commutators of the fractional integral operator and the related maximal operator are established.     

Multiple orthogonal polynomials associated with the exponential integral

We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights $(w_1,w_2)$ on the positive real line, with $w_1(x)=x^{\alpha }{e}^{-x}$ the gamma density and $w_2(x)=x^{\alpha }{E}_{\nu +1}(x)$ a density related to the exponential integral $E_{\nu +1}$. We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series, and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider two related families of mixed-type multiple orthogonal polynomials.     

Weighted estimates for fractional integral operators on central Morrey spaces

We prove weighted estimates for fractional integral operators on central Morrey spaces. Our result covers the weighted theorem by De Napoli, Drelichman and Durán (2011). Our proof is different from theirs.     

Asymptotic type for sectorial operators and an integral of fractional powers

There is a standard notion of type for a sectorial linear operator acting in a Banach space. We introduce a notion of asymptotic type for a linear operator V, involving estimates on the resolvent ⁻¹(λI+V) as λ→0. We show, for example, that if V is sectorial and of asymptotic type ω then the fractional power Vα is of asymptotic type αω for a suitable range of positive α. Moreover, we establish various properties of the operator ; in particular, this operator is of asymptotic type 0, for a sectorial operator V. This result has an application to the construction of operators satisfying the well-known Ritt resolvent condition.     

Bounds for singular fractional integrals and related Fourier integral operators

We prove sharp L^p−L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature hypothesis.